# Tagged Questions

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### What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$ $\lfloor\,\cdot\,\rfloor$ represents the greatest integer

Question : What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$. $\lfloor\,\cdot\,\rfloor$ represents the greatest integer function less than or equal to $x$. I know how to ...
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### Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
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### Creating a function with logarithmic growth

I have some knobs with an internal value of $0$ to $1$. These represent a value in a range, like $1$ to $1000$. Case in point, I would like to be able to change the scale/growth of the display value. ...
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### graphing $f(x)=x \ln \left(1+\frac{1}{x}\right)$

I was assigned to draw the graph of this function $f(x)$=$x\ln(1+{1\over x})$. And when I calculate $\lim_{x\to \infty} f(x)$ I get $1$ but the teacher it's not correct even though its graph on the ...
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### For $0<a<b$, show $1-\dfrac{a}{b} < \log\left( \dfrac{b}{a} \right) < \dfrac{b}{a}-1$

Prove that if, $0 < a < b$ Then $1-\dfrac{a}{b} < \log\left( \dfrac{b}{a} \right) < \dfrac{b}{a}-1$
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### Show that $\operatorname{ln}(n!)=\Theta(n\operatorname{ln}(n))$

Another question about asymptotic approximations. We are asked to show that $\operatorname{ln}(n!)=\Theta(n\operatorname{ln}(n))$ I'm stuck tho and can use help. What I did is: ...
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### check my short simple proof - Functions are of same magnitude. Asymptotic notation.

A simple question with a short solution I thought of, but I would like verification. $f(n)$ is a function that approaches infinity as $n$ approaches infinity. We are asked to show that ...
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### Allowed values for $x$ in $\log_2(x)$

$$y=\log_2x$$ What are the allowed values for $x$ in this function? How do I calculate it? (I know how it works for normal functions with fractions and other stuff, but this one I'm stuck)
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### Writing an equation for a log function given the graph

I have the following graph for a logarithmic function $f$: I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph ...
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### Minimum value of $f(x) = x\log_2x +(1-x)\log_2(1-x)$ [closed]

What is the minimum value of the following function for $0<x<1$ ? Here the base of logarithm is 2 . $f(x) = x\log_2x +(1-x)\log_2(1-x)$
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### Why isn't $\log(-1)$ defined?

Why isn't $\log(-1)$ defined. It can be defined as being equal to $i\pi$. Why don't we define the $log$ function over Complex Numbers as well.
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### function symmetric around a point

I need some quick help solving this: What is y(ln(2))if the function y satisfies $$\frac{dy}{dx}=1-y^2$$ and is symmetric about the point (ln(4),0)? I know that a function is symmetric about the ...
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### Use a graph to estimate the time at which the number was increasing most rapidly

For the period from 2000 to 2008, the percentage of households in a certain country with at least one DVD player has been modeled by the function $f(t) = \frac{87.5}{1 + 17.1e^{−0.91t}}$ where the ...
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### At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
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### Expressing logarithms as ratios of natural logarithms

$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an ...
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### Understanding a question on iterated logarithms

I have in front of me a math problem that I do not understand. That's to say, I don't understand what is being asked of me. Problem: We can define $\log_2**(x) = log_2*(log_2*(x))$ and the function ...
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### Making logarithmic function go higher

I am looking at logarithmic functions, and, lets say, log2 (x+3) is having a bit of a growth rate between 0-10 values of ...
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### Online logarithm drawing

I am looking for a site that will give me the output of my logarithms. What I want to do, is I want to input, in example log(2), and I want it to draw an output ...
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### domain of function with logarithmic terms.

what will be the domain of function given below? $$y=1+3(\log(\sin(x))+\log(\csc(x)))$$ in book it is given this is valid for the values of angles of 1st and 2nd quadrant only. why this function is ...
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### Find the inverse function…

So, I have the function $$f(x)=\frac{2^x-2^{-x}}{2}.$$ I tried finding the inverse function the usual way I do, but I guess I'm stuck with these degrees. So far, I've come to this form ...
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### Filling in 'x' in a log function

if $3^5=x$ (exponential equation) converts to log form gives $log_3x=5$ that makes sense. $$3^5 = 243 \Rightarrow x=243$$ So if I take the log form again: $log_3x=5$ and replace $x$ with $243$. I ...
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### How do I solve such logarithm

I understand that $\log_b n = x \iff b^x = n$ But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$) What if I don't? For example, how do I solve for $x$ ...
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### determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
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### What's the most straight forward way to show that a function is increasing?

I am trying to show that: $$\frac{2}{n}\log\Gamma\left(\frac{x}{2}\right) - \log\Gamma\left(\frac{x+n-1}{n}\right)$$ is an increasing function for $x \ge 5$ and $n > 2$ One way to do this would ...
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### Comparing rates of change: which function increases faster?

I am comparing two functions for $x \ge 1$: $$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$ g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...